Positive solutions for singular fractional differential equations with three-point boundary conditions

Abstract

In this paper, we consider the nonlinear three-point boundary value problem of singular fractional differential equations

$$\begin{aligned} D^{\alpha }_{0^+}u(t)+a(t)f(t,u(t))=0,\quad 0<t<1,\;2<\alpha \le 3 \end{aligned}$$

with boundary conditions

$$\begin{aligned}u(0)=0,\quad D^{\beta }_{0^+}u(0)=0,\quad D^{\beta }_{0^{+}}u(1)=bD^{\beta }_{0^{+}}u(\xi ),\quad 1\le \beta \le 2 \end{aligned}$$

involving Riemann–Liouville fractional derivatives $D^{\alpha }_{0^{+}}$ and $D^{\beta }_{0^{+}}$. The nonlinear term f permits singularities with respect to both the time and space variables. We obtain several local existence and multiplicity of positive solutions theorems by introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions. An example is given to show the applicability of our main results.